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Let $P \subset \mathbb R^n$ be a Delzant polytope defined by inequalities $\ell_i(x) \geq 0, i=1, \ldots, d$.

Of course, from the symplectic point of view, the inequalities $a_i \ell_i \geq 0$ still define the same polytope for all positive real $a_i$.

What I wonder is the following. Does the function $g: P \rightarrow \mathbb R$ (up to affine terms) $$ g(x) = \sum_{i=1}^d (a_i \ell_i(x) \log \ell_i(x)) + h(x) $$ with $h$ smooth on $P$ define a compatible complex structure on $P$, no matter what the $a_i>0$ are?

Or are legal potentials only the ones where the $\ell_i(x)$ are of the form $\langle x, \mu_i \rangle - \lambda_i$ where $\mu_i$ is a primitive element of the integral lattice and points to the interior of the $i$-th face?

thank you David

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Yes, but it might be singular. For instance, if $a_i = 1/\beta_i$, then the potential $u$ corresponds to a conical K\"ahler metric with cone angles $2\pi\beta_i$ along the divisor corresponding to the face $[l_i=0]$. In particular if you take $a_i$ to be an integer it will correspond to an orbifold structure.

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thanks! Can you share some references about this? –  David Petrecca Feb 1 at 8:23
There is a paper of Lerman and Tolman that says that there is one-one correspondence between toric orbifolds and Delzant polytopes with integer weights attached to the faces. arxiv.org/abs/dg-ga/9412005 The conic case generalizes this to the situation where weights are allowed to be non-integer. I dont know if there is a proof of this in literature, but it seems to be well known. At least in many of Donaldson's writings on the subject it seems implicit. –  Ved Datar Feb 4 at 18:28
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